1. The Ubiquitous MATCHED FILTER . . . it’s everywhere!!
an evening with a very important
principle that’s finding exciting
new applications in modern radar
R. T. Hill AES Society
an IEEE Lecturer Dallas Chapter
25 September 2007

2. Wow! . . all in twenty minutes or so!
How do receivers work?
A brief review, then, of
network theory
characterizing a receiver by its
“impulse response function”
representing radio signals
reminding ourselves of “convolution”
in linear systems . .
Wow! . . all in twenty minutes or so!
Well, first we’ll need a receiver block diagram 

3. Generalized Receiver Block Diagram

4. “characterizing” a receiver (or generally, a network)
by its “impulse response function”
Why?

5. Do you see how we can represent any signal s(t) as a
collection of impulses . .?
. . the impulse is a wonderful function, so useful –
I’ll make some comments about it.

6. Also, our signal (certainly in radio/radar work)
is surely bipolar-cyclic . . a modulated carrier
. . that is, we can think of it, in radio work, as a sine wave
of voltage field intensity and polarity, leaving all the relationships
to its accompanying magnetic field and the medium at hand
“to Maxwell”, so to speak!
Do you see now
how impulses
could still be used
to represent even
this complex radio
signal?

7. +
Now, we see here that the output is indeed the sum
of the many impulse response functions, weighted and
translated by the input signal . . that the action of this
linear network is, by superposition, a convolution!

8. About designing our receiver . .
What impulse response function
do I want??
Well, what do we want our
receiver to do . . produce a
replica of our signal? NO!!
. . . discuss . . .

9. Oh, yes . . did I neglect to mention “noise”? 
For maximum sensitivity to our own signal’s being at
the input, we don’t need to see a copy of it at the output . .
. . we simply need, at the output, the greatest possible
indication of its presence at the input.
In other words, we need to have an impulse response
function that, when convolved with our signal, would
give the greatest possible “signal to noise ratio” at the
output.
Oh, yes . . did I neglect to mention “noise”? 

10. Yes, noise . . always present in radio work . .
Admitting, then, that our total input is, say, “gi(t)”,
made up of our desired signal AND accompanying
(but wholly independent) noise, we see that we can
treat these two inputs separately as they pass through
our receiver:
Yes, noise . . always present in radio work . .
Oh, yes , , looks complicated, but nothing new here . . just
remember the words here! Let the math “talk” to you!
and our output is just the sum of the two convolutions – one due
to the signal being present and one from the always-present noise.

11. So, we might ask . . Answer . . . . WHAT Impulse Response Function
would produce the greatest possible indication, at the
output, of our signal’s arrival at the input?
Answer . .
. . various methods (differential calculus; the “Schwartz
inequality”) lead to the conclusion that maximum sensitivity
is achieved when the IRF is the complex conjugate of the
subject signal:
Concepts:
► Signal to Noise Ratio (SNR) as a measure of sensitivity
► Representing our cyclic signal in this “A ejθ “ form
and, note that some details of necessary time
displacement notation are overlooked here 
So, we might ask . .

12. The Matched Filter This, then, is
A receiver the impulse response function of which is
the complex conjugate of a particular signal will produce
the greatest possible signal-to-noise ratio at the output
when that signal is at the input and in the presence of
independent and completely random noise this
receiver is the most “sensitive” to the particular signal . .
. . it is “matched” to it.

13. Representing our signal by a rotating vector . .
. . a great convenience
. . a vector rotating at the carrier frequency, amplitude modulated
by s(t) with possible phase modulation (a binary phase coded signal,
for example) shown as Φ(t) do you know, or remember, this
convention? Sure helps in diagramming a lot of things in today’s
signal processing.

14. Some discussion . . to improve our understanding
. . consider a four-segment amplitude- and phase-modulated signal,
and for the moment, without noise . .
Some discussion . . to improve our understanding
. . something similar to a child’s
scattering the blocks with which he had
made a tower . . what if we wanted to
see that maximum height (rebuild the
tower) again? I would re-align the blocks
by multiplying each by its conjugate
(remember: angles add when vectors
are multiplied) and – voila! – the tower
appears again, maximum possible height!
Is “angle” then enough? The conjugate involves the amplitude . .why?

15. In conjugating the phase modulation of our signal, why multiply
in the convolution by the amplitude as well?
Ah . . we cannot ignore the noise!
The circles here show an expectation of the noise contribution.
Our input signal gi(t) is our own signal abcd and this noise . . but
notice, the noise is (of course) of the same strength regardless of
the amplitude of abcd at that time. Also note, the noise is completely
random. This utter randomness and independence of abcd are properties
of “Gaussian” noise, “white” noise, as from natural thermal phenomena.
Now, to convolve with, say, a unit level phase-only conjugate would
exaggerate some of the noise effects in the angle-corrected vector
addition – unwise. The best thing for us to do, in such noise, is
indeed to ignore it! “Match” to our signal alone!

16. OK . . but just one further thought . .
Remember the child’s block tower? Consider:
The child’s playroom is subject to a mild earthquake
(good grief!) as the blocks are tumbled in the way
we expected.
“White” noise . . we’re OK . . use the matched filter
On the other hand, what if a “wind” had been blowing
distinctly from, say, the west as the blocks were tumbled
in addition to the earthquake’s vibratory behavior?
Not random!! Biased! We’d better compensate
for that, assuming we can sense it. That is,
we may wish to use a “whitening” filter to
randomize the disturbance before the matched filter!
That idea is indeed “key” to much of the adaptive signal processing
so strong in today’s radar literature more about that to come

17. Where do we use the Matched Filter? Some illustrations . .
Pulse compression, very common in radar . .
Receive beam steering, direction finding in antennas,
the “adaptive antenna” . .
Space-Time Adaptive Processing (STAP) in radar . .
Polarimetry in radar, adaptive processing, target recognition . .

18. Illustration # 1 . . Pulse Compression
► First, some remarks about pulse compression
in modern radar . . fine range resolution desired,
but still with long pulse for lots of energy
► Achieved by modulating the transmitted pulse,
then “compressing” the pulse on receive
with (of course) a matched filter
► Techniques – binary phase coding widely used;
typical lengths of hundreds to one – our
example here? A mere four to one! 

19.   + + - + Binary phase coding with a “tapped delay line”
first bit out
Binary phase coding
with a “tapped delay line”
Showing a 180o
phase shift on one tap,
giving the binary code sequence “ “ , one of the Barker codes. 
On receive (after down conversion), the signal is sent through
its Matched Filter . . in this case, the same circuit with the taps reversed:
Clearly, pulse compression is a “convolution” process, and we see the “time” or “range” sidelobes
in the output which, for all the Barker binary codes, are never more than unity value, while the narrow
main peak is full value, the number of bits in the code. In this matched situation, the output is the
“autocorrelation function”, and a low sidelobe level is a very desirable attribute of a candidate code.

20. A peculiar thing about binary phase coding
The idea that the “tapped delay line, backwards”
is indeed a conjugating matched filter is not so clear
in binary phase coding . . adding or subtracting 180°
results in the same zero phase for that bit (all bits
then being phase aligned).
Just to illustrate conjugation more clearly, imagine that our
four-segment sequence had been 0°, -30°, 0°, 0° – terrible
autocorrelation function, but it makes our point about phase
“realignment” by conjugation in this convolution process.
Pulse expander
on transmit
Pulse compressor
on receive
Compressed pulse output (here
showing rather poor range sidelobes)

21. The Barker codes: sidelobe level Length 2 + - and + + - 6.0 dB
and dB
Modulo 2 adder
Seven-stage shift register
This seven-stage shift register is used to generate a 127-bit binary sequence
that can in turn be used to control a (0,180o) phase shifter through which our
IF signal is passed in the pulse modulator of our waveform generator. Such
shift-register generators produce sequences of length 2N – 1 (before repeating;
N is the number of stages). Today, computer programs generate the modulation,
storing sequences known to have good autocorrelation functions, for many lengths
other than just 2Int - 1.

22. Illustration # 2 . . Antennas; the “Adaptive” Array
► First, some remarks about how antennas form receive
beams, phased arrays the simplest and very
pertinent illustration
► Next, we’ll observe that compensating for the “angle
of arrival” for an echo is indeed a form of (you guessed it)
a matched filter, this one in “angle space”
► Then, we’ll consider (again in angle space) that the “noise”
may NOT be utterly random, not “white” (statistically
uniform) in angle . . we may need a “whitening” filter
before our matched filter 

23. First, consider a few discrete elements of a phased array, a line array . .
These simple sketches
will remind us of how
antennas, phased arrays
specifically, perform beam
steering . . a matter of
compensating for the
element-to-element phase
difference resulting from
the path-length differences
associated with the desired
beam-steering angle (the
angle-of-arrival “under test”,
so to speak) 

24. . . continuing . . Can you see how the “compensating” phase
control in the phased
array is acting as a
conjugating matched
filter? Here, the “segments”
of our “signal” are NOT
a function of time (as before
discussed in signal processing)
but rather a function of space,
the position of each element
of the array. The same Matched Filter principle applies, but here in
a different “dimensional space” than normally considered in matched
filter teaching.
But what was the other part of the principle?
Ah . . that the “background noise” be independent and random . .
. . necessary condition for the Matched Filter to give
best possible output (here, angle measurement accuracy).
Is our noise stationary in angle, uniform statistically??? Perhaps not!! 

25. The “Adaptive Antenna” . .
Discussion
► Spatial analysis analogous
to “spectral analysis”
► Finding compensating weights
for each element involves
solving as many simultaneous
algebraic equations . . inverting
the covariance matrix NOT EASY
► Adapted pattern will be “inverse” to the
angular distribution of noise, “whitening” it
► Today’s art state DOF rather standard . .
Array signal processing . . first, spatial analysis,
then compensation to “whiten”
Bottom line – the coherent sidelobe cancellers (CSLC)
– the more elaborate “adaptive phased arrays”
are forms of spatial “whitening filters”, here to “whiten”
the heterogeneous disturbance – noise – in angle. Why? So that
a straightforward angle matched filter can be used most effectively.

26. Illustration # 3 . . Space-Time Adaptive Processing, STAP
► Adaptive antenna processing is Space-Adaptive. What is
meant by Space-Time Adaptive Processing?
► A few remarks about Doppler processing in radar, itself
an application of the Matched Filter . . Doppler filters
are indeed filters matched to a particular Doppler shift
► Many radars, airborne ones particularly, need to do Doppler
processing when the background (noise, continuous ground
clutter) is certainly NOT spectrally uniform . . once again,
we’ll need a “whitening” filter

27. Doppler filtering
Theory View a single Doppler “filter” as a classic “Matched Filter”,
that is, we multiply (convolve) the input signal
with the conjugate of the signal being sought.
sample #
signal x
reference =
product
Recall, phase angles add when complex numbers (vectors) are multiplied – that is,
the signal is “rotated back” in phase by the amount it might have been progressing
in phase . . To the extent that such a component was in the input signal will we get
an output in this particular filter. We’ve built a Matched Filter for that component
(that frequency component) alone. HOWEVER, this is best ONLY IF background
noise is utterly random in Doppler frequencies 

28. The airborne radar situation . . for discussion
Do you see the need for “whitening” the background
in BOTH the angle dimension (the broadband interference
is purely angle dependent) and in Doppler shift (the ground
reflectivity may NOT be utterly random, uniformly distributed in angle)?
Broadband interference
(jamming) suggests need
for adaptive antenna
Terrain features
contribute to
non-uniform spectrum
of the side-lobe coupled
ground clutter
An airborne radar

29. and also in the weights to put on each pulse return to shape the
STAP – to be adaptive in both the antenna’s pattern (as before discussed)
and also in the weights to put on each pulse return to shape the
Doppler filters in spectrum (compensating for the non-uniform
spectrum of the background clutter)
The “data field”
available to us
To adapt to the background’s sensed heterogeneity in both angle and spectrum,
we must solve (to be “fully adaptive”) MN simultaneous equations (size of the covariance
matrix to invert: MN x MN. No wonder, then, today’s literature is full of STAP papers
addressing ways to “reduce the dimensionality” of the processing, find the best that
we can do in “partial adaptivity”! Very exciting work!
Space
Time

30. Illustration # 4 . . The Polarimetric Matched Filter
► First, a short general review of polarimetry in radar,
its uses, its value
► Then, an example of the Polarimetric Whitening Filter
and how a polarimetric radar image (by SAR)
is improved just from PWF application to the area clutter
► Of course, the “whitening” to randomize the polarization state
of the surrounding area (local clutter in a scene) permits us
then to search for targets (building, vehicles) the
“polarimetric signature” of which may have been
estimated in advance.

31. Radar Polarimetry . . a little review
► Polarization of an Electro-Magnetic wave is taken as the
spatial orientation of the E-field . . most, but certainly
not all, radars are designed to operate, for various
reasons, in either horizontal or vertical (linear)
polarization, fixed by the antenna design – that is,
they are not “polarimetric”
► A “Fully Polarimetric Radar” (FPR) can, first, transmit one
polarization and separately measure the received
signal in each of two orthogonal polarizations, then
do the same, transmitting the orthogonal polarization
(e.g., transmit H, receive H and V;
then transmit V, receive H and V)
► We learn a lot about a target by sensing its polarimetric scattering
► Developed well by the meteorological radar community, some
other specialty radars

32.  Polarimetry used for image enhancement
“Whitening” and “Matching” filters
► The work under Dr. Les Novak (MIT/Lincoln Laboratories) in the 1990s is extremely
valuable in establishing these approaches to image enhancement by polarimetry.
A number of papers in our conferences (to be cited here) and other teaching material
he has provided me contribute to this instruction. An airborne SAR at 33 GHz,
fully polarimetric, was used in many valuable experiments there.
► Review . . Detection of things of interest (targets) in the presence of return
not of interest (noise, clutter) requires contrast between the two in some
observable dimension space (here, our image).
► The idea of “whitening” and “matching” is universal, forms matched filter theory.
● The whitening filter: attempts to minimize the “speckle” of the background,
– that is, the standard deviation among the pixels of the clutter – in
images formed by combining the complex images in HH, HV and VV
using complex weights among them, weights that minimize the
correlation in cluttered regions among the three images. The weights
are based on knowledge of the clutter covariance matrix, a priori in the
Novak work reviewed here, and involved its inversion, not difficult with
order three. With such PWF, cells that do not “belong” to the clutter will
have increased contrast with the background and are more easily seen.
● The matched filter: attempts to maximize the target intensity to clutter intensity
in the combined image, by using weights based on knowledge (estimates)
of the polarimetric covariance of target AND clutter returns. 

33. . . from the Novak, Lincoln Laboratory work on
(Polarimetry and image enhancement, cont.)
. . from the Novak, Lincoln Laboratory work on
PWF . . a dual-power-line scene, images by
the 33 GHz fully polarimetric airborne SAR.
The histograms show the increased contrast
(separation of the clutter and towers compilations)
afforded by PWF processing compared to a
non-polarimetric image, here the HH.
One can see the visible effect in the two images
above, HH on the left, PWF-processed at right.
All here with 1 foot x 1 foot resolution.

34. Well . . did we make it to this concluding slide?? The Matched Filter
● The conjugate impulse response function –
max sensitivity to a signal
in the presence of white noise
● Normally taught in the context of just temporal signal
processing . . functions of time, etc
● Should be no less seen by students of radar as the
underlying principle to many advances, in “other” dimension spaces: angle (antenna
patterns), spectral analysis (Doppler filtering),
polarimetric analysis (as in synthetic aperture
radar image enhancement)
● Today’s “adaptive” processes are generally the MF-related
“whitening” required in non-random environments

35. More than you wanted to know about
The End
More than you wanted to know about
The Matched Filter